Solve exactly for

x: sin^3 x + sin x cos^2 x = cos x

If 0 degrees <= x <= 180 degrees

x = ? degrees?

Would i factor out sin^3 x from the left side and the right. and try to solve from there?

never mind disregard this. I found the answer to be 45 degrees

sin^3 x + sinx cos^2 x = cosx

sinx(sin^2 x + cos^2 x) = cosx
sinx (1) = cosx
sinx/cosx = 1
tanx = 1
x = 45° or x = 225°
but you wanted to be between 0 and 180

so x = 45°

To solve the equation sin^3(x) + sin(x) * cos^2(x) = cos(x), you can follow these steps:

1. Factor out sin(x) from the left side of the equation:
sin(x) * (sin^2(x) + cos^2(x)) + sin(x) * cos^2(x) = cos(x)

2. Simplify the equation:
sin(x) + sin(x) * cos^2(x) = cos(x)

3. Subtract cos(x) from both sides of the equation:
sin(x) + sin(x) * cos^2(x) - cos(x) = 0

4. Factor out sin(x) from the left side of the equation:
sin(x) * (1 + cos^2(x)) - cos(x) = 0

5. Expand the equation:
sin(x) + sin(x) * cos^2(x) - cos(x) = 0

6. Rearrange the terms:
sin(x) * (1 + cos^2(x)) - cos(x) = 0

At this point, factoring out sin^3(x) may not lead to a simple solution. Let's try solving the equation using an alternative method.

7. Rewrite the equation using trigonometric identities:
sin(x) * (sin^2(x) + cos^2(x)) - cos(x) = 0
sin(x) * 1 - cos(x) = 0
sin(x) - cos(x) = 0

8. Square both sides of the equation:
(sin(x) - cos(x))^2 = 0^2
sin^2(x) - 2 * sin(x) * cos(x) + cos^2(x) = 0
1 - 2 * sin(x) * cos(x) = 0

9. Simplify the equation:
1 = 2 * sin(x) * cos(x)

10. Divide both sides of the equation by 2:
0.5 = sin(x) * cos(x)

Now, since 0 degrees <= x <= 180 degrees, we can analyze the possible values.

Possible scenarios:
a) sin(x) = 0.5 and cos(x) = 1
b) sin(x) = 1 and cos(x) = 0.5

In scenario a), sin(x) = 0.5, which occurs at angles x = 30 degrees and x = 150 degrees. However, cos(x) = 1, which does not satisfy the equation.

In scenario b), cos(x) = 0.5, which occurs at angles x = 60 degrees and x = 120 degrees. In this case, sin(x) = 1, which satisfies the equation.

Therefore, the solutions for x are x = 60 degrees and x = 120 degrees.